{
  "id": "1206.5949",
  "version": "v3",
  "published": "2012-06-26T11:06:07.000Z",
  "updated": "2013-01-12T03:04:23.000Z",
  "title": "On the topology of free paratopological groups. II",
  "authors": [
    "Ali Sayed Elfard",
    "Peter Nickolas"
  ],
  "comment": "This paper has been published by Topology and its Applications",
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $\\FP(X)$ be the free paratopological group on a topological space $X$. For $n\\in \\N$, denote by $\\FP_n(X)$ the subset of $\\FP(X)$ consisting of all words of reduced length at most $n$, and by $i_n$ the natural mapping from $(X\\oplus X^{-1}\\oplus \\{e\\})^n$ to $\\FP_n(X)$. In this paper a neighbourhood base at the identity $e$ in $\\FP_2(X)$ is found. A number of characterisations are then given of the circumstances under which $i_2\\colon (X\\oplus X^{-1}_d\\oplus \\{e\\})^2\\to \\FP_2(X)$ is a quotient map, where $X$ is a $T_1$ space and $X^{-1}_d$ denotes the set $X^{-1}$ equipped with the discrete topology. Further characterisations are given in the case where $X$ is a transitive $T_1$ space. Several specific spaces and classes of spaces are also examined. For example, $i_2$ is a quotient for every countable subspace of $\\R$, $i_2$ is not a quotient for any uncountable compact subspace of $\\R$, and it is undecidable in ZFC whether an uncountable subspace of $\\R$ exists for which $i_2$ is a quotient.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-01-12T03:04:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free paratopological group",
      "uncountable compact subspace",
      "specific spaces",
      "discrete topology",
      "quotient map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.5949S"
    }
  }
}